Dynamical behavior of differential equation models of frequency and density dependent populations

This paper studies the dynamics of a mathematical model of a continuously reproducing diploid population with two alleles at one locus. The dependent variables are allele frequency and population density. If the genotype fitnesses are frequency and density dependent, the stability of equilibria is related to the geometry of the zero allele fitness curves. The asymptotic behavior of solutions where fitness is only density dependent is contrasted to the asymptotic behavior where fitness is frequency and density dependent. A parameterized family of fitness functions giving a Hopf bifurcation and limit cycles is investigated analytically and numerically.     

The effect of index selection on allele frequencies and future genetic gains when traits are correlated

The application of the selection index in the case of an additive two-trait model in which the genetic effect on each trait is determined by a finite number of loci is examined. Simulation results indicate that the direction of change in the frequency of favourable alleles is not necessarily in the positive direction at all loci when index selection is used as the basis for truncation selection. When the genetic correlation was positive (or favourable with respect to the economic weights), there was little difference (<5%) in genetic gain over 20 generations and no difference in the direction of change in allele frequencies or genetic correlation whether or not updated values for the genetic (co)variances were used in constructing the selection index. However, when the genetic correlation was negative or unfavourable, the effect of using genetic parameters which were not updated had unexpected effects on the allele frequencies and genetic correlation and reduced the genetic gain by a greater amount (< 12%).     

Coexistence of competing species by the oscillation of polymorphisms

Scale-eating cichlids in Lake Tanganyika exhibit genetically determined lateral asymmetry, especially in their mouth-opening. Frequencies of the morphs oscillate due to strong frequency-dependent selection caused by the switching of prey's attention, and its delayed effect by their growth period. Two scale-eaters coexist in similar densities at south shore of the lake, with their morph frequencies oscillating in phase. We investigated the effect of the oscillation in morph frequencies to the coexistence of competing species. If the difference of two species' growth period is large, the oscillation facilitates the coexistence of the two species, while small difference of growth periods hinders their coexistence. In the latter case, the species with shorter growth period drives the other species to the extinction.     

Coexistence of competing species of seaweed flies: the role of temperature

Abstract.

• 1 Competition is shown to be occurring within and between the congeners Coelopa frigida and C.pilipes; it is noted that the two species frequently coexist despite ongoing competition.
• 2 Observations on natural wrack-beds indicate that there is a marked difference in the distributions of the larvae: C.frigida larvae aggregate in cooler parts of the bed, C.pilipes in warmer parts. This difference in microdistribution reflects a broader-scale difference in the geographical distributions of the species - C.pilipes being the more southerly of the two.
• 3 The larval distributions are shown to be caused primarily by the behaviour of the larvae themselves - not by choices made by ovipositing females, nor (at least to any great extent) by differential survival.
• 4 The different micro-distributions within beds constitute a form of niche difference which will cause competitive abilities to be frequency-dependent and hence have a stabilizing effect. It is possible that this effect may be supplemented by others; and, in particular, effects operating at the level of the‘linear meta-population’may be worthy of further investigation both in Coelopa and in other coastal species.

     

Disturbances allow coexistence of competing species

We give the first mathematically rigorous proof that disturbances allow competing species to coexist. This work provides a mathematical framework to explain the existence of fugitive species and the role played by disturbances in increasing or decreasing the biodiversity of ecosystems. We study modifications of the metapopulation model for patchy environments proposed by Caswell and Cohen (1990, 1991). For the one- and two-species models we give necessary and sufficient conditions on the parameters for the existence of a non-trivial equilibrium solution, which is shown to be always globally stable.     

A quantitative genetic model for analyzing species differences in outcrossing species

A genetic model based on a two-level intra- and interspecific mating design is proposed to estimate the genetic architecture of species differences and heterosis for outcrossing species. The underlying genetic analyses make use of classical quantitative genetic theories and recent results from molecular genetic studies. Gene effects across different quantitative trait loci (QTL) can be approximated by a geometric series. Under natural selection, gene effects are often associated with allele frequencies in a particular way, which can be approximated by the gamma distribution. By incorporating these approximations into family structural analyses in the mating design, we are able to estimate a number of genetic parameters that contribute to quantitative genetic variation based on a nonlinear optimization approach. These parameters include the number of QTL, their gene effects, and their allele frequencies in the parental populations. We perform simulation studies and illustrate an example to demonstrate the statistical property and procedure of the method.     

Dynamical behavior for population genetics models of differential and difference equations with nonmonotone fitnesses

This paper studies the dynamical behavior of classical 2-dimensional models of continuously and discretely reproducing diploid populations with two alleles at one locus. The phase variables are allele frequency and population density. The genotype fitnesses are not assumed to be monotonically decreasing functions of density. Hence the mean fitness curve is more complicated than in the monotonic case. If genotype fitnesses are only density dependent, results concerning equilibrium stability are obtained similar to those for the monotonic case, and periodic solutions are precluded in the differential equation model. An example with one-hump genotype fitnesses is presented and analyzed.Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

Spatial patterns of coexistence of competing species in patchy habitat

The single-species spatially realistic patch occupancy metapopulation model is, in this study, extended to a metacommunity of many competing species. Competition is assumed to reduce the local carrying capacity (effective patch area), which in turn increases local extinction rates and reduces colonization rates because of smaller population sizes. Each species is described by three parameters: pre-competitive abundance (equilibrium incidence of patch occupancy, which reflects the rate of colonization in relation to extinction rate), the spatial range of migration, and competitive ability. The model ignores spatio–temporal correlations caused by interspecific interactions, because in metacommunities of unequal competitors inhabiting heterogeneous landscapes, correlations in the occurrence of species are driven more by patch heterogeneity than by competition. The model allows the calculation of multispecies equilibria in patchy habitats without simulations. In general, the number of coexisting species in the metacommunity increases with decreasing strength of competition, increasing rate of colonization, and decreasing range of migration. Habitat heterogeneity in the form of spatial variation in patch areas tends to facilitate coexistence. Poor competitors may coexist with superior competitors in the patch network if the former have higher colonization rates (competition–colonization trade-off). When migration distances are short, competition leads to spatial pattern formation: Species tend to have restricted spatial distributions in the network, but contrary to intuitive expectations, often the distributions of many species are nested. Having more dispersive species enhances both local and global diversity, whereas more local migration decreases local but increases global diversity.     

Fixation in haploid populations exhibiting density dependence II: the quasi-neutral case

We determine fixation probabilities in a model of two competing types with density dependence. The model is defined as a two-dimensional birth-and-death process with density-independent death rates, and birth rates that are a linearly decreasing function of total population density. We treat the 'quasi-neutral case' where both types have the same equilibrium population densities. This condition results in birth rates that are proportional to death rates. This can be viewed as a life history trade-off. The deterministic dynamics possesses a stable manifold of mixtures of the two types. We show that the fixation probability is asymptotically equal to the fixation probability at the point where the deterministic flow intersects this manifold. The deterministic dynamics predicts an increase in the proportion of the type with higher birth rate in growing populations (and a decrease in shrinking populations). Growing (shrinking) populations therefore intersect the manifold at a higher (lower) than initial proportion of this type. On the center manifold, the fixation probability is a quadratic function of initial proportion, with a disadvantage to the type with higher birth rate. This disadvantage arises from the larger fluctuations in population density for this type. These results are asymptotically exact and have relevance for allele fixation, models of species abundance, and epidemiological models.     

The effects of ecological and genetic neighbourhood size on the evolution of two competing species

Summary Individual-based simulations were conducted to examine the effect of a small ecological neighbourhood (an area in which ecological processes such as density-dependent factors operate) and the genetic neighbourhood size (the size of an area from which the parents may be assumed to be drawn at random) on the coevolution of two competing species. For the simulations, individuals of two consumer species compete for two types of food organisms. Different genotypes (one locus and two alleles) have different efficiencies of food acquisition for different food types. Individual consumer organisms search for food within their home ranges and reproduce depending on the amount of food eaten. The dispersal distance of the offspring follows a normal distribution with a zero mean and _d standard deviation. Simulations were conducted by varying the home range size, mating area (area from where individuals choose their mates), standard deviation of dispersal distance, food generation time, the reproductive rates of food populations and the sizes and number of independent food populations. Food organisms reproduce either within one population or independently within 16 spatially divided populations. For all the simulations, competitive exclusion was the most frequent outcome and character displacement was the least frequent outcome. Through a 200-generation simulation, the two consumer species could co-exist longer and maintain a polymorphic resource use longer when the home range and mating size were small in 16 spatially divided populations than when random mating and homogeneous interaction occurred within a community (perfect mixing population). For perfect mixing populations, the frequency of character displacement increased as the food generation time became short and the reproductive rates of food decreased. It follows from the results that the sizes of the genetic and ecological neighbourhoods and the mode of resource dynamics can affect the evolution of two competing species.     

Development and characterization of 14 novel microsatellite markers from the golden pheasant (Chrysolophus pictus)

Fourteen polymorphic microsatellite loci were isolated from the golden pheasant (Chrysolophus pictus). The average allele number of these microsatellites was 11.14 per locus, ranging from two to seventeen in a group of 31 individuals. The mean observed and expected heterozygosities were 0.763 and 0.722, respectively. The average polymorphic information content value, cumulative discrimination power, overall probability of exclusion with both unknown parents or with only unknown sire for these markers were 0.741, 1.0, 0.9999, and 1.0, respectively. This set of microsatellite markers would provide useful tools for conservation genetic studies of the golden pheasant.     

The effect of Streptomyces species on the survival of Salmonella in soil

Abstract A modified Chelex 100 ion-exchange extraction method was used to monitor streptomycete spores, streptomycete mycelia and Salmonella in soil. Salmonella dusseldorf in soil was inhibited by the bactericidal effect of streptomycin and by the growth of Streptomyces bikiniensis . The soil used in the experiments exerted an antimicrobial effect on S. dusseldorf .
Competition between S. dusseldorf, Streptomyces lividans TK24 and Stm. bikiniensis ATCC 11062 was monitored in soil. In sterile amended soil Stm. lividans increased the survival of S. dusseldorf , whereas survival was reduced in the presence of the known streptomycin producer, Stm. bikiniensis . In the presence of S. dusseldorf the production of spores and mycelia by Stm. bikiniensis was reduced, and Stm. lividans sporulation was reduced but mycelia production increased. Evidence was seen for a beneficial effect between S. dusseldorf and Stm. lividans mycelia.
In non-sterile unamended soil the survival of S. dusseldorf was increased in the presence of Stm. lividans , whereas Stm. bikiniensis had no effect. Stm. lividans and Stm. bikiniensis reduced the survival of S. dusseldorf in non-sterile amended soil, with the most dramatic reduction caused by Stm. bikiniensis . No such changes in the survival of S. dusseldorf were observed with non-sterile amended soil that had been treated with sludge. The presence of sludge in unamended soil increased the rate of Salmonella die-off. In unamended soil containing sludge the presence of Stm. lividans increased the survival of S. dusseldorf , whereas survival was reduced in the presence of Stm. bikiniensis . The data provided evidence of antibiosis in soil, relating to the possible production of streptomycin by Stm. bikiniensis .     

Effects of variation in individual growth on plant species coexistence

Abstract. Both size structure and variability (spatial heterogeneity, disturbance, stochasticity, variation in species attributes, etc.) are regarded as regulatory mechanisms of species coexistence. However, none of the models so far proposed consider both size structure and variability simultaneously. A size-structured variation model for plant-community dynamics is proposed, which is based on the diffusion model for growth dynamics of plant populations. This model has four functions: (1) mean growth rate of individuals of size x at time t, G(t, x) (species-specific mean traits, e.g. competitive ability); (2) variance in growth rate of individuals of size x at time t, D(t, x) (stochastic factors due to genetic variation, environmental heterogeneity, spatial variation of individuals, etc.); (3) mortality rate of individuals of size x at time t, M(t, x); and (4) recruitment rate at time t, R(t), as a boundary condition. The interference function for individuals of size x at time t, C(t, x), is introduced, which expresses the degree of interactions between individuals and hence averaged effects of local neighbourhood competition; the G(t, x), D(t, x), M(t, x) and R(t) functions are given in terms of C(t, x). These four functions describe the growth dynamics of individuals of each species in the plant community. Effects of the G(t, x), D(t, x), M(t, x) and R(t) functions on species coexistence in plant communities were evaluated by simulation and the relative importance of the D(t, x) function as well as size structure was shown for species coexistence especially in plant communities where competition among species is non-transitive or niche limitation does not work.     

Methodologies for estimating the sample size required for genetic conservation of outbreeding crops

Summary The main purpose of germplasm banks is to preserve the genetic variability existing in crop species. The effectiveness of the regeneration of collections stored in gene banks is affected by factors such as sample size, random genetic drift, and seed viability. The objective of this paper is to review probability models and population genetics theory to determine the choice of sample size used for seed regeneration. A number of conclusions can be drawn from the results. First, the size of the sample depends largely on the frequency of the least common allele or genotype. Genotypes or alleles occurring at frequencies of more than 10% can be preserved with a sample size of 40 individuals. A sample size of 100 individuals will preserve genotypes (alleles) that occur at frequencies of 5%. If the frequency of rare genotypes (alleles) drops below 5%, larger sample sizes are required. A second conclusion is that for two, three, and four alleles per locus the sample size required to include a copy of each allele depends more on the frequency of the rare allele or alleles than on the number. Samples of 300 to 400 are required to preserve alleles that are present at a frequency of 1%. Third, if seed is bulked, the expected number of parents involved in any sample drawn from the bulk will be less than the number of parents included in the bulk. Fourth, to maintain a rate of breeding (F) of 1 %, the effective population size (N _e) should be at least 150 for three alleles, and 300 for four alleles. Fifth, equalizing the reproductive output of each family to two progeny doubles the effective size of the population. Based on the results presented here, a practical option is considered for regenerating maize seed in a program constrained by limited funds.Part of this paper was presented at the Global Maize Germplasm Workshop, CIMMYT, El Batan, Mexico, March 6–12, 1988     

Genetics and the environment in interspecific competition: a study using the sibling species Drosophila melanogaster and Drosophila simulans

The outcome of interspecific competition of two closely related species may depend upon genetic variation in the two species and the environment in which the experiment is carried out. Interspecific competition in the two sibling species, Drosophila melanogaster and D. simulans, is usually investigated using longterm laboratory stocks that often have mutant markers that distinguish them. To examine competition in flies that genetically more closely resemble flies in nature, we utilized freshly caught wildtype isofemale lines of the two species collected at the same site in San Carlos, Mexico. Under ordinary laboratory conditions, D. melanogaster always won in competition. However, in hotter and drier conditions, D. simulans competed much more effectively. In these environmental conditions, there were genetic differences in competitive ability among lines with the outcome of competition primarily dependent upon the line of D. melanogaster used but in some cases also influenced by the line of D. simulans used. Differences in the measures of productivity and developmental time did not explain the differences in competitive ability among lines. This suggests that the outcome of competition was not due to differences in major fitness components among the isofemale lines but to some other attribute(s) that influenced competitive ability. When lines of flies were combined, the outcome of competition was generally consistent with competitive outcomes between pairs of lines. In several cases, the combination of lines performed better than the best of the constituent lines, suggesting that competitive ability was combined heterotically and that the total amount of genetic variation was important in the outcome of interspecific competition.     

The rules of variation: Amino acid exchange according to the rotating circular genetic code

General guidelines for the molecular basis of functional variation are presented while focused on the rotating circular genetic code and allowable exchanges that make it resistant to genetic diseases under normal conditions. The rules of variation, bioinformatics aids for preventative medicine, are: (1) same position in the four quadrants for hydrophobic codons, (2) same or contiguous position in two quadrants for synonymous or related codons, and (3) same quadrant for equivalent codons. To preserve protein function, amino acid exchange according to the first rule takes into account the positional homology of essential hydrophobic amino acids with every codon with a central uracil in the four quadrants, the second rule includes codons for identical, acidic, or their amidic amino acids present in two quadrants, and the third rule, the smaller, aromatic, stop codons, and basic amino acids, each in proximity within a 90 degree angle. I also define codifying genes and palindromati, CTCGTGCCGAATTCGGCACGAG.     

Roles of maternal effects in maintaining genetic variation: Maternal storage effect

Understanding the maintenance of genetic variation remains a central challenge in evolutionary biology. Recent empirical studies suggest the importance of temporally varying selection, as allele frequencies have been found to fluctuate substantially in the wild. However, previous theory suggests that the conditions for the maintenance of genetic variation under temporally fluctuating selection are quite restrictive. Using mathematical models, we demonstrate that maternal genetic effects, whereby maternal genotypes affect offspring phenotypes, can facilitate the maintenance of polymorphism in temporally varying environments. Maternal effects result in mismatches between genotypes and phenotypes, thereby buffering the influence of selection on allele frequency. This decreases the magnitude of allele‐frequency fluctuations and creates conditions for the maintenance of variation when selection causes fluctuations. Therefore, maternal effects may result in a temporal storage effect (“maternal storage effect”). On the other hand, when selection does not cause fluctuations (e.g., linear negative frequency‐dependent selection), maternal genetic effects moderate the relative importance of selection compared to genetic drift and promote stochastic allele extinction in finite populations. Thus, maternal effects can play an important role in the maintenance of polymorphism, but the direction of the effect depends on the nature of selection.     

Multilocus patterns of genetic variation across Cryptosporidium species suggest balancing selection at the gp60 locus

Cryptosporidium is an apicomplexan protozoan that lives in most vertebrates, including humans. Its gp60 gene is functionally involved in its attachment to host cells, and its high level of genetic variation has made it the reference marker for sample typing in epidemiological studies. To understand the origin of such high diversity and to determine the extent to which this classification applies to the rest of the genome, we analysed the patterns of variation at gp60 and nine other nuclear loci in isolates of three Cryptosporidium species. Most loci showed low genetic polymorphism (π_S <1%) and similar levels of between‐species divergence. Contrastingly, gp60 exhibited very different characteristics: (i) it was nearly ten times more variable than the other loci; (ii) it displayed a significant excess of polymorphisms relative to between‐species differences in a maximum‐likelihood Hudson–Kreitman–Aguadé test; (iii) gp60 subtypes turned out to be much older than the species they were found in; and (iv) showed a significant excess of polymorphic variants shared across species from random expectations. These observations suggest that this locus evolves under balancing selection and specifically under negative frequency‐dependent selection (FDS). Interestingly, genetic variation at the other loci clusters very well within the groups of isolates defined by gp60 subtypes, which may provide new tools to understand the genome‐wide patterns of genetic variation of the parasite in the wild. These results suggest that gp60 plays an active and essential role in the life cycle of the parasite and that genetic variation at this locus might be essential for the parasite's long‐term success.     

Mathematical properties of the r2 measure of linkage disequilibrium

Statistics for linkage disequilibrium (LD), the non-random association of alleles at two loci, depend on the frequencies of the alleles at the loci under consideration. Here, we examine the r(2) measure of LD and its mathematical relationship to allele frequencies, quantifying the constraints on its maximum value. Assuming independent uniform distributions for the allele frequencies of two biallelic loci, we find that the mean maximum value of r(2) is approximately 0.43051, and that r(2) can exceed a threshold of 4/5 in only approximately 14.232% of the allele frequency space. If one locus is assumed to have known allele frequencies--the situation in an association study in which LD between a known marker locus and an unknown trait locus is of interest--we find that the mean maximum value of r(2) is greatest when the known locus has a minor allele frequency of approximately 0.30131. We find that in 1/4 of the space of allowed values of minor allele frequencies and haplotype frequencies at a pair of loci, the unconstrained maximum r(2) allowing for the possibility of recombination between the loci exceeds the constrained maximum assuming that no recombination has occurred. Finally, we use r(max)(2) to examine the connection between r(2) and the D(') measure of linkage disequilibrium, finding that r(2)/r(max)(2)=D('2) for approximately 72.683% of the space of allowed values of (p(a),p(b),p(ab)). Our results concerning the properties of r(2) have the potential to inform the interpretation of unusual LD behavior and to assist in the design of LD-based association-mapping studies.     

The effect of epistasis on sexually antagonistic genetic variation

There is increasing evidence of segregating sexually antagonistic (SA) genetic variation for fitness in laboratory and wild populations, yet the conditions for the maintenance of such variation can be restrictive. Epistatic interactions between genes can contribute to the maintenance of genetic variance in fitness and we suggest that epistasis between SA genes should be pervasive. Here, we explore its effect on SA genetic variation in fitness using a two locus model with negative epistasis. Our results demonstrate that epistasis often increases the parameter space showing polymorphism for SA loci. This is because selection in one locus is affected by allele frequencies at the other, which can act to balance net selection in males and females. Increased linkage between SA loci had more marginal effects. We also show that under some conditions, large portions of the parameter space evolve to a state where male benefit alleles are fixed at one locus and female benefit alleles at the other. This novel effect of epistasis on SA loci, which we term the ‘equity effect’, may have important effects on population differentiation and may contribute to speciation. More generally, these results support the suggestion that epistasis contributes to population divergence.